Discrete Time Signal and Down Sampling

A discrete time signal is a sequence of values that are defined at discrete points in time. In this case, the given discrete time signal is represented by x(n) = e^(j(5π/3)) * e^(j(π/4)), where e represents the base of the natural logarithm and j is the imaginary unit.

Down sampling is the process of reducing the number of samples in a signal by selecting every nth sample, where n is an integer. In this case, the signal x(n) is down sampled to xd(n) = x(4n).

Fundamental Period of Down Sampled Signal

To determine the fundamental period of the down sampled signal xd(n), we need to find the value of n for which xd(n) repeats itself.

Let's analyze the down sampled signal xd(n) = x(4n) in terms of its samples:

- When n = 0, xd(0) = x(0) = e^(j(5π/3)) * e^(j(π/4))
- When n = 1, xd(1) = x(4) = e^(j(5π/3)) * e^(j(π/4))
- When n = 2, xd(2) = x(8) = e^(j(5π/3)) * e^(j(π/4))
- When n = 3, xd(3) = x(12) = e^(j(5π/3)) * e^(j(π/4))
- ...

From the above samples, we can observe that xd(0) = xd(1) = xd(2) = xd(3) = ... = e^(j(5π/3)) * e^(j(π/4)). Therefore, the down sampled signal xd(n) repeats itself after every 4 samples.

Fundamental Period Calculation

The fundamental period of a signal is the smallest positive integer that represents the period of the signal. In this case, the down sampled signal xd(n) has a period of 4 samples.

Hence, the fundamental period of the down sampled signal xd(n) is 4.

Summary

- The given discrete time signal x(n) = e^(j(5π/3)) * e^(j(π/4)) is down sampled to xd(n) = x(4n).
- The down sampled signal xd(n) repeats itself after every 4 samples, indicating a period of 4.
- Therefore, the fundamental period of the down sampled signal xd(n) is 4.